Question

Liliana solved the equation below. When she tried to verify her answer, she realized she made an error.

One-third (x + 18) = 7
One-third x + 6 = 7
One-third x + 6 minus 6 = 7 + 6
One-third x = 1
x = 3

Which correction should Liliana make?
The first step should be One-third x + 54 = 7.
The second step should be One-third x + 6 minus 6 = 7 minus 6.
The third step should be One-third x = 13.
The final step should be x = one-third.

Answer 1

Let's analyze Liliana's original equation and her steps to find where she went wrong:

The equation is:
\[
\frac{1}{3}(x + 18) = 7
\]

1. **Correct first step**: Liliana's assertion that "One-third (x + 18) = 7" is correct. However, her next step should be to distribute the \(\frac{1}{3}\) through the parentheses:
\[
\frac{1}{3}x + 6 = 7
\]
This matches her second step, so this part is correct.

2. **Incorrect arithmetic in the second step**:
Here, she attempts to subtract 6 from both sides, but she writes:
\[
\frac{1}{3}x + 6 \text{ minus } 6 = 7 + 6
\]
This is incorrect. The correct operation should be:
\[
\frac{1}{3}x + 6 - 6 = 7 - 6.
\]
So:
\[
\frac{1}{3}x = 1.
\]

3. **Computing the value of x**: Now, using the correct equation:
\[
\frac{1}{3}x = 1
\]
To isolate \(x\), we multiply both sides by 3:
\[
x = 3.
\]

Thus, her final step was correct in obtaining \(x = 3\); however, since she misapplied the subtraction, the relevant correction she needs to make is:

**The second step should be "One-third x + 6 minus 6 = 7 minus 6."**

Regarding the listed options, the appropriate choice for correction is:

**The second step should be One-third x + 6 minus 6 = 7 minus 6.**